direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C11×Dic6, C33⋊4Q8, C44.3S3, C12.1C22, C132.5C2, C22.13D6, Dic3.C22, C66.18C22, C3⋊(Q8×C11), C4.(S3×C11), C2.3(S3×C22), C6.1(C2×C22), (C11×Dic3).2C2, SmallGroup(264,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×Dic6
G = < a,b,c | a11=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×Dic6
►Regular action on 264 points
Generators in S264
(1 106 183 214 60 90 141 162 169 48 264)(2 107 184 215 49 91 142 163 170 37 253)(3 108 185 216 50 92 143 164 171 38 254)(4 97 186 205 51 93 144 165 172 39 255)(5 98 187 206 52 94 133 166 173 40 256)(6 99 188 207 53 95 134 167 174 41 257)(7 100 189 208 54 96 135 168 175 42 258)(8 101 190 209 55 85 136 157 176 43 259)(9 102 191 210 56 86 137 158 177 44 260)(10 103 192 211 57 87 138 159 178 45 261)(11 104 181 212 58 88 139 160 179 46 262)(12 105 182 213 59 89 140 161 180 47 263)(13 31 239 68 110 250 151 79 130 222 196)(14 32 240 69 111 251 152 80 131 223 197)(15 33 229 70 112 252 153 81 132 224 198)(16 34 230 71 113 241 154 82 121 225 199)(17 35 231 72 114 242 155 83 122 226 200)(18 36 232 61 115 243 156 84 123 227 201)(19 25 233 62 116 244 145 73 124 228 202)(20 26 234 63 117 245 146 74 125 217 203)(21 27 235 64 118 246 147 75 126 218 204)(22 28 236 65 119 247 148 76 127 219 193)(23 29 237 66 120 248 149 77 128 220 194)(24 30 238 67 109 249 150 78 129 221 195)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204)(205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264)
(1 222 7 228)(2 221 8 227)(3 220 9 226)(4 219 10 225)(5 218 11 224)(6 217 12 223)(13 189 19 183)(14 188 20 182)(15 187 21 181)(16 186 22 192)(17 185 23 191)(18 184 24 190)(25 214 31 208)(26 213 32 207)(27 212 33 206)(28 211 34 205)(29 210 35 216)(30 209 36 215)(37 78 43 84)(38 77 44 83)(39 76 45 82)(40 75 46 81)(41 74 47 80)(42 73 48 79)(49 238 55 232)(50 237 56 231)(51 236 57 230)(52 235 58 229)(53 234 59 240)(54 233 60 239)(61 91 67 85)(62 90 68 96)(63 89 69 95)(64 88 70 94)(65 87 71 93)(66 86 72 92)(97 193 103 199)(98 204 104 198)(99 203 105 197)(100 202 106 196)(101 201 107 195)(102 200 108 194)(109 136 115 142)(110 135 116 141)(111 134 117 140)(112 133 118 139)(113 144 119 138)(114 143 120 137)(121 255 127 261)(122 254 128 260)(123 253 129 259)(124 264 130 258)(125 263 131 257)(126 262 132 256)(145 169 151 175)(146 180 152 174)(147 179 153 173)(148 178 154 172)(149 177 155 171)(150 176 156 170)(157 243 163 249)(158 242 164 248)(159 241 165 247)(160 252 166 246)(161 251 167 245)(162 250 168 244)
G:=sub<Sym(264)| (1,106,183,214,60,90,141,162,169,48,264)(2,107,184,215,49,91,142,163,170,37,253)(3,108,185,216,50,92,143,164,171,38,254)(4,97,186,205,51,93,144,165,172,39,255)(5,98,187,206,52,94,133,166,173,40,256)(6,99,188,207,53,95,134,167,174,41,257)(7,100,189,208,54,96,135,168,175,42,258)(8,101,190,209,55,85,136,157,176,43,259)(9,102,191,210,56,86,137,158,177,44,260)(10,103,192,211,57,87,138,159,178,45,261)(11,104,181,212,58,88,139,160,179,46,262)(12,105,182,213,59,89,140,161,180,47,263)(13,31,239,68,110,250,151,79,130,222,196)(14,32,240,69,111,251,152,80,131,223,197)(15,33,229,70,112,252,153,81,132,224,198)(16,34,230,71,113,241,154,82,121,225,199)(17,35,231,72,114,242,155,83,122,226,200)(18,36,232,61,115,243,156,84,123,227,201)(19,25,233,62,116,244,145,73,124,228,202)(20,26,234,63,117,245,146,74,125,217,203)(21,27,235,64,118,246,147,75,126,218,204)(22,28,236,65,119,247,148,76,127,219,193)(23,29,237,66,120,248,149,77,128,220,194)(24,30,238,67,109,249,150,78,129,221,195), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264), (1,222,7,228)(2,221,8,227)(3,220,9,226)(4,219,10,225)(5,218,11,224)(6,217,12,223)(13,189,19,183)(14,188,20,182)(15,187,21,181)(16,186,22,192)(17,185,23,191)(18,184,24,190)(25,214,31,208)(26,213,32,207)(27,212,33,206)(28,211,34,205)(29,210,35,216)(30,209,36,215)(37,78,43,84)(38,77,44,83)(39,76,45,82)(40,75,46,81)(41,74,47,80)(42,73,48,79)(49,238,55,232)(50,237,56,231)(51,236,57,230)(52,235,58,229)(53,234,59,240)(54,233,60,239)(61,91,67,85)(62,90,68,96)(63,89,69,95)(64,88,70,94)(65,87,71,93)(66,86,72,92)(97,193,103,199)(98,204,104,198)(99,203,105,197)(100,202,106,196)(101,201,107,195)(102,200,108,194)(109,136,115,142)(110,135,116,141)(111,134,117,140)(112,133,118,139)(113,144,119,138)(114,143,120,137)(121,255,127,261)(122,254,128,260)(123,253,129,259)(124,264,130,258)(125,263,131,257)(126,262,132,256)(145,169,151,175)(146,180,152,174)(147,179,153,173)(148,178,154,172)(149,177,155,171)(150,176,156,170)(157,243,163,249)(158,242,164,248)(159,241,165,247)(160,252,166,246)(161,251,167,245)(162,250,168,244)>;
G:=Group( (1,106,183,214,60,90,141,162,169,48,264)(2,107,184,215,49,91,142,163,170,37,253)(3,108,185,216,50,92,143,164,171,38,254)(4,97,186,205,51,93,144,165,172,39,255)(5,98,187,206,52,94,133,166,173,40,256)(6,99,188,207,53,95,134,167,174,41,257)(7,100,189,208,54,96,135,168,175,42,258)(8,101,190,209,55,85,136,157,176,43,259)(9,102,191,210,56,86,137,158,177,44,260)(10,103,192,211,57,87,138,159,178,45,261)(11,104,181,212,58,88,139,160,179,46,262)(12,105,182,213,59,89,140,161,180,47,263)(13,31,239,68,110,250,151,79,130,222,196)(14,32,240,69,111,251,152,80,131,223,197)(15,33,229,70,112,252,153,81,132,224,198)(16,34,230,71,113,241,154,82,121,225,199)(17,35,231,72,114,242,155,83,122,226,200)(18,36,232,61,115,243,156,84,123,227,201)(19,25,233,62,116,244,145,73,124,228,202)(20,26,234,63,117,245,146,74,125,217,203)(21,27,235,64,118,246,147,75,126,218,204)(22,28,236,65,119,247,148,76,127,219,193)(23,29,237,66,120,248,149,77,128,220,194)(24,30,238,67,109,249,150,78,129,221,195), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264), (1,222,7,228)(2,221,8,227)(3,220,9,226)(4,219,10,225)(5,218,11,224)(6,217,12,223)(13,189,19,183)(14,188,20,182)(15,187,21,181)(16,186,22,192)(17,185,23,191)(18,184,24,190)(25,214,31,208)(26,213,32,207)(27,212,33,206)(28,211,34,205)(29,210,35,216)(30,209,36,215)(37,78,43,84)(38,77,44,83)(39,76,45,82)(40,75,46,81)(41,74,47,80)(42,73,48,79)(49,238,55,232)(50,237,56,231)(51,236,57,230)(52,235,58,229)(53,234,59,240)(54,233,60,239)(61,91,67,85)(62,90,68,96)(63,89,69,95)(64,88,70,94)(65,87,71,93)(66,86,72,92)(97,193,103,199)(98,204,104,198)(99,203,105,197)(100,202,106,196)(101,201,107,195)(102,200,108,194)(109,136,115,142)(110,135,116,141)(111,134,117,140)(112,133,118,139)(113,144,119,138)(114,143,120,137)(121,255,127,261)(122,254,128,260)(123,253,129,259)(124,264,130,258)(125,263,131,257)(126,262,132,256)(145,169,151,175)(146,180,152,174)(147,179,153,173)(148,178,154,172)(149,177,155,171)(150,176,156,170)(157,243,163,249)(158,242,164,248)(159,241,165,247)(160,252,166,246)(161,251,167,245)(162,250,168,244) );
G=PermutationGroup([[(1,106,183,214,60,90,141,162,169,48,264),(2,107,184,215,49,91,142,163,170,37,253),(3,108,185,216,50,92,143,164,171,38,254),(4,97,186,205,51,93,144,165,172,39,255),(5,98,187,206,52,94,133,166,173,40,256),(6,99,188,207,53,95,134,167,174,41,257),(7,100,189,208,54,96,135,168,175,42,258),(8,101,190,209,55,85,136,157,176,43,259),(9,102,191,210,56,86,137,158,177,44,260),(10,103,192,211,57,87,138,159,178,45,261),(11,104,181,212,58,88,139,160,179,46,262),(12,105,182,213,59,89,140,161,180,47,263),(13,31,239,68,110,250,151,79,130,222,196),(14,32,240,69,111,251,152,80,131,223,197),(15,33,229,70,112,252,153,81,132,224,198),(16,34,230,71,113,241,154,82,121,225,199),(17,35,231,72,114,242,155,83,122,226,200),(18,36,232,61,115,243,156,84,123,227,201),(19,25,233,62,116,244,145,73,124,228,202),(20,26,234,63,117,245,146,74,125,217,203),(21,27,235,64,118,246,147,75,126,218,204),(22,28,236,65,119,247,148,76,127,219,193),(23,29,237,66,120,248,149,77,128,220,194),(24,30,238,67,109,249,150,78,129,221,195)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204),(205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228),(229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264)], [(1,222,7,228),(2,221,8,227),(3,220,9,226),(4,219,10,225),(5,218,11,224),(6,217,12,223),(13,189,19,183),(14,188,20,182),(15,187,21,181),(16,186,22,192),(17,185,23,191),(18,184,24,190),(25,214,31,208),(26,213,32,207),(27,212,33,206),(28,211,34,205),(29,210,35,216),(30,209,36,215),(37,78,43,84),(38,77,44,83),(39,76,45,82),(40,75,46,81),(41,74,47,80),(42,73,48,79),(49,238,55,232),(50,237,56,231),(51,236,57,230),(52,235,58,229),(53,234,59,240),(54,233,60,239),(61,91,67,85),(62,90,68,96),(63,89,69,95),(64,88,70,94),(65,87,71,93),(66,86,72,92),(97,193,103,199),(98,204,104,198),(99,203,105,197),(100,202,106,196),(101,201,107,195),(102,200,108,194),(109,136,115,142),(110,135,116,141),(111,134,117,140),(112,133,118,139),(113,144,119,138),(114,143,120,137),(121,255,127,261),(122,254,128,260),(123,253,129,259),(124,264,130,258),(125,263,131,257),(126,262,132,256),(145,169,151,175),(146,180,152,174),(147,179,153,173),(148,178,154,172),(149,177,155,171),(150,176,156,170),(157,243,163,249),(158,242,164,248),(159,241,165,247),(160,252,166,246),(161,251,167,245),(162,250,168,244)]])
99 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 11A | ··· | 11J | 12A | 12B | 22A | ··· | 22J | 33A | ··· | 33J | 44A | ··· | 44J | 44K | ··· | 44AD | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | S3 | Q8 | D6 | Dic6 | S3×C11 | Q8×C11 | S3×C22 | C11×Dic6 |
| kernel | C11×Dic6 | C11×Dic3 | C132 | Dic6 | Dic3 | C12 | C44 | C33 | C22 | C11 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 1 | 1 | 1 | 2 | 10 | 10 | 10 | 20 |
Matrix representation of C11×Dic6 ►in GL2(𝔽23) generated by
| 12 | 0 |
| 0 | 12 |
| 0 | 9 |
| 5 | 16 |
| 15 | 17 |
| 7 | 8 |
G:=sub<GL(2,GF(23))| [12,0,0,12],[0,5,9,16],[15,7,17,8] >;
C11×Dic6 in GAP, Magma, Sage, TeX
C_{11}\times {\rm Dic}_6 % in TeX
G:=Group("C11xDic6"); // GroupNames label
G:=SmallGroup(264,18);
// by ID
G=gap.SmallGroup(264,18);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-3,220,461,226,4404]);
// Polycyclic
G:=Group<a,b,c|a^11=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export